Symmetries of geometric objects play crucial role in mathematics, physics, and computer science. The mathematical approach uses group actions: symmetries of objects form a group, and the primary goal is to parametrise families of objects up to symmetry.

These parameter spaces–called moduli spaces–often arise as quotients of phase spaces by group actions, and the complexity of moduli spaces fundamentally depends on the structure of the symmetry group. David Mumford in 1974 received the Fields medal (the Nobel prize in mathematics) for his celebrated Geometric Invariant Theory (GIT), which gives a systematic description of quotients of algebraic varieties by reductive algebraic groups – a fundamental theory which has become a basic tool for mathematicians and physicists in the last 60 years. This theory, however, fails at a fundamental level when the acting group is non-reductive. The proposed project aims to combine recent major developments in global singularity theory with new pioneering techniques in Non-Reductive Geometric Invariant Theory (NRGIT) to tackle some open problems on Hilbert scheme of points on manifolds, which are mathematical models for configuration spaces of points (point datasets).